Continuity of Parameterized Optimal Solution

Question

Suppose for every $y$, $f(x,y)$ is strictly convex in $x$. Further, $f(x,y)$ is continuous in $y$. Let $\mathcal X$ be compact (in my problem, $\mathcal X$ is an interval). Can anyone suggest any theorems or references for the problem of whether $x^*(y) = \text{argmin}_{x \in \mathcal X} f(x,y)$ is continuous in $y$?

Answer 1

It is not necessarily continuous. For example, the function $f(x,y)=x^2,\, y\leq 0$ and $f(x,y)=(x-1)^2,\,y>0$, where $x\in [-2,2]$, $y\in[-1,1]$.

You can see that if $\,y\to 0^+\quad x^*(y)\to 1$ and if $\,y\to 0^-\quad x^*(y)\to 0$, because $x^*(y)=0,\,y\leq 0$ and $x^*(y)=1,\,y>0$